We consider quantum many-body dynamics under quantum measurements, where the measurement-induced phase transitions (MIPs) occur when changing the frequency of the measurement. In this work, we consider the robustness of the MIP for long-range interaction that decays as r-a with distance r. The effects of long-range interactions are classified into two regimes: (i) the MIP is observed (a>ac), and (ii) the MIP is absent even for arbitrarily strong measurements (a<ac). Using fermion models, we demonstrate both regimes in integrable and nonintegrable cases. We identify the underlying mechanism and propose sufficient conditions to observe the MIP, that is, a>d/2+1 for general bilinear systems and a>d+1 for general nonintegrable systems (d: spatial dimension). Numerical calculation indicates that these conditions are optimal.
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